The simple pendulum

For a simple pendulum, the theory (and the intuition) shows that two equilibrium points exist. One is stable (q=0 and w=0) and the other one is unstable (q=p and w=0).

The phase graph obtained with the numeric calculation is the following :

It corresponds to the theory.

Additional comments

The mass m of the pendulum does not appear anymore in the equation of motion. Galileo Galilei (1564-1642) was the first who discovered this effect. Maybe you know his legendary experiment of dropping two balls of the same size but of different mass from the tower of Pisa, where both balls had reached the ground simultaneously. From the viewpoint of Newton's laws, there is no reason that the inertial mass has to be the same as the gravitational mass. It was the ingenious idea of Albert Einstein (1879-1955) to take this equality not as accidentally but as a deep principle of nature. From this equivalence of gravitational and inertial mass, he developed a new understanding of gravity which led to his general theory of relativity.